Asymptotic and Non-asymptotic Estimates for Multivariate Laplace

Inequalities and Asymptotic Estimates

Unless specified otherwise, we use µ, σ 2 to denote the mean and variance of the the variable under consideration. This note shall be updated throughout the seminar as I find more useful inequalities. 1 Basic inequalities Theorem 1.1 (Markov's Inequality). If X is a random variable taking only non-negative values, then for any a > 0 Pr[X ≥ a] ≤ E[X] a. (1) Proof. We show this for the discrete c...

Laplace Asymptotic Expansions for Gaussian Functional Integrals

We obtain a Laplace asymptotic expansion, in orders of λ, of Eρx { G(λx)e −2F (λx)} } the expectation being with respect to a Gaussian process. We extend a result of Pincus [9] and build upon the previous work of Davies and Truman [1, 2, 3, 4]. Our methods differ from those of Ellis and Rosen [6, 7, 8] in that we use the supremum norm to simplify the application of the result. A.M.S. Classifica...

Asymptotic Behavior of Multivariate Reward Processes with Nonlinear Reward Functions

Asymptotic Laplace Transforms and Evolution Equations

Some non asymptotic tail estimates for Hawkes processes

We use the Poisson cluster process structure of a Hawkes process to derive non asymptotic estimates of the tail of the extinction time, of the coupling time or of the number of points per interval. This allows us to define a family of independent Hawkes processes ; each of them approximating the initial process on a particular interval. Then we can easily derive exponential inequalities for Haw...